Line segment detection using weighted mean shift procedures on a 2D slice sampling strategy

THEORETICAL ADVANCES

Line segment detection using weighted mean shift procedureson a 2D slice sampling strategy

Marcos Nieto • Carlos Cuevas • Luis Salgado •

Narciso Garcıa

Received: 4 August 2009 / Accepted: 26 March 2011 / Published online: 9 April 2011

� Springer-Verlag London Limited 2011

Abstract A new line segment detection approach is

introduced in this paper for its application in real-time

computer vision systems. It has been designed to work

unsupervised without any prior knowledge of the imaged

scene; hence, it does not require tuning of input parameters.

Although many works have been presented on this topic, as

far as we know, none of them achieves a trade-off between

accuracy and speed as our strategy does. The reduction of

the computational cost compared to other fast methods is

based on a very efficient sampling strategy that sequen-

tially proposes points on the image that likely belong to

line segments. Then, a fast line growing algorithm is

applied based on the Bresenham algorithm, which is

combined with a modified version of the mean shift algo-

rithm to provide accurate line segments while being robust

against noise. The performance of this strategy is tested for

a wide variety of images, comparing its results with pop-

ular state-of-the-art line segment detection methods. The

results show that our proposal outperforms these works

considering simultaneously accuracy in the results and

processing speed.

Keywords Line segment � Eigenvalues � Real time � Slice

sampling �Mean shift � Bresenham algorithm

1 Introduction

Straight lines or line segments within an imaged scene can

be of great help to infer its geometric properties and also

the projection process that converts 3D world elements into

a 2D image. Man-made environments typically contain

multiple line segments oriented towards a number of

common directions (belonging to flat surfaces such as the

ground, doors, walls, or buildings). For that reason, line

segments can be used as low level features for conventional

computer vision problems, such as the detection of van-

ishing points [2, 20], autocalibration [27], plane rectifica-

tion techniques [21, 23], 3D reconstruction [22], object

tracking [29] or active countours tracking [30].

This paper models the problem of finding line segments in

real images in a probabilistic way. An image is considered as a

set of observations I ¼ fðxk; ykÞgk¼1...N coming from a sub-

jacent probability density function p(x, y) that represents the

probability of a pixel to belong to a line segment. One of the

main contributions of the paper is on the generation of an

estimate pðx; yÞ of this likelihood distribution from the anal-

ysis of the eigenvalues of the tensor matrix associated to the

image pixels. Besides, the real-time performance of the pro-

posed strategy is achieved by the use of a sequential sampling

technique that selects pixels in the image that likely belong to

line segments according to pðx; yÞ: Line segments are

obtained through the Bresenham growing algorithm enhanced

with the use of weighted mean shift (wMS) procedures that

provides accurate fits and robustness against noise.

The target of our method is the obtention of line seg-

ments in any kind of real images. Therefore, Sect. 7

M. Nieto (&)

Vicomtech-ik4, Mikeletegi Pasealekua 57,

20009 Donostia-San Sebastian, Spain

e-mail: [email protected]

C. Cuevas � L. Salgado � N. Garcıa

Grupo de Tratamiento de Imagenes,

Universidad Politecnica de Madrid, 28040 Madrid, Spain


L. Salgado


N. Garcıa


123

Pattern Anal Applic (2011) 14:149–163

DOI 10.1007/s10044-011-0211-4

evaluates the obtained line segments for a wide variety of

real images. Nevertheless, since one of the major feats of

our approach is its low computational cost, its performance

is better exploited for vision applications working on

sequences of images. For instance, camera autocalibration

strategies based on the computation of vanishing points in

moving camera sequences of structured environments [20]

require real-time operation for accurate and meaningful

line segment detections.

2 Related work

Most approaches in the related literature use edges

extracted at pixel level from the images as basic informa-

tion. The Sobel edge detector [16] is typically used as an

approximation to the first order spatial derivatives Ix(x, y)

and Iy(x, y). For each pixel, k at location (xk, yk) the module

and orientation of the gradient can also be estimated as

described in [9].

Once obtained, the information of edges at pixel level

can be used in different ways to search for line segments.

Two main groups of techniques can be identified: those

based on accumulation in parametric spaces, such as

methods based on the Hough transform (HT) [1, 3, 9, 11,

17, 19, 31]; and pixel clustering strategies in the image

plane [6, 13, 14, 18].

2.1 Hough transform

The former group basically performs a parametric trans-

formation to the set of edge pixels, from which lines are

extracted searching for local maxima in the transform

domain. Note that the standard HT (SHT) detects lines, but

not line segments, which require addressing additional

considerations for their detection, as done in [18, 35]. Other

works face the problem of finding lines, as done by [19, 31,

34] or recently Aggarwal [1] with the Regularized HT.

The major drawback of the works based on the HT is the

excessive algorithm complexity [9, 31], which avoids its

use on online applications.

Random sampling applied to the HT (RHT) has been

originally proposed by Xu et al. [34], and further improved

by Kiryati et al. [19] to render efficient line segment

detectors. A pair of edge points is selected at random to

accumulate a single vote on the transformed domain. These

approaches are the so-called ‘‘many-to-one’’ voting

schemes, in contrast to the ‘‘one-to-many’’ corresponding

to the standard HT, which dramatically enhances the

speediness of the HT. Several variants of the RHT have

been proposed by Kalviainen et al. [17], which improve the

sampling distribution by means of a two-step random

process. Windows of fixed or also random size are

randomly selected in the image, and the RHT is then

applied to search for maxima. Walsh and Raftery [32]

proposed an importance sampling procedure to improve the

random sampling used within the RHT.

The resulting line segments can be further grouped into

longer ones as done by Bandera et al. [3], which uses mean

shift procedures to cluster line segments in the transform

domain and obtain a single representative for each cluster.

Nevertheless, the main problem of these approaches is

that they achieve fast results at the cost of reducing their

accuracy, giving a large number of false negatives (miss-

detections due to the termination criteria), especially in

very fast detectors as the PPHT (Progressive Probabilistic

HT) by Matas et al. [11]. Besides, they require a large set

of application-dependant threshold values to be tuned to

obtain adequate results. As final remark, the need of such a

number of user-defined thresholds in this type of approa-

ches reduces its applicability to unsupervised systems as

well as making them more prone to errors.

2.2 Pixel clustering

This group first cluster sets of connected pixels in a coarse

manner according to some common property (e.g. their

orientation) and afterwards compute an estimate for each

set. The work by Burns et al. [6] has been used as reference

by further authors [13, 18]. Besides, the use of connected

component analysis (CCA) was first introduced in this

context by Nevatia and Babu [26] and followed by other

authors [18, 20]. Analogously, Yuen et al. [35] propose a

connected version of the HT, which selects at random an

edge pixel and then apply an one-dimensional accumula-

tion on the angle parameter.

In this field, some of the works exploit the information

contained in the eigenvalues and vectors of the image

tensor matrix [33], as done by many others [14, 18, 20].

Typically, the straightness of the group of pixels is mea-

sured as a function on the eigenvalues (k1, k2). However, it

is not straightforward to classify a pixel, given its eigen-

values, into these three categories. As mentioned by Rosten

et al. [28], many authors have created unbounded functions

that evaluate the ‘‘cornerness’’ of pixels by the computation

of k2/k1, k2, or approximations to the relationship between

eigenvalues that skip their computation using directly the

elements of the tensor matrix, such as the well-known

Harris corner function.

3 Approach overview

The proposed strategy gathers properties of the two

reviewed groups of line detection methods, although it

cannot be classified as belonging to any of them. On the

150 Pattern Anal Applic (2011) 14:149–163

123

one hand, it uses a sequential sampling strategy, which is

an improved version of the random sampling approaches

used by Hough-based methods. On the other hand, the

proposed method uses the information of the second

moment matrix at pixel level as also done by some clus-

tering-based methods, although in a novel way, since it

computes a new likelihood function pðx; yÞ:The flow chart of the proposed method (which will be

denoted as SSWMS, slice sampling weighted mean shift),

is depicted in Fig. 1. As shown, it is composed of three

main stages: the computation of the likelihood function, the

sequential sampling, and the line segment generation.

The first step estimates the likelihood distribution over

the image I such that pðx; yÞ is the likelihood value of a

pixel (x, y) to belong to a line segment. This distribution is

parameterized by (l1, l2), which are statistics automati-

cally obtained from the image. The distribution is defined

using the image tensor matrix, and their corresponding

eigenvalues. This process is guided by the idea that pixels

that belong to line segments must satisfy two criteria: (1)

they have significant gradient magnitude; and (2) there is

only one dominant direction in their neighborhoods. These

concepts and the associated methods are described in detail

in further sections.

The sequential sampling, based on the slice sampling

algorithm [4, 25], sequentially selects pixels in the image,

denoted as zk ¼ ðxk; ykÞ; that are good candidates to belong

to line segments, according to the computed likelihood

pðzkÞ: The design of this stage ensures that no repeated

samples are selected, so that zk 6¼ zc8c\k: If the slice

sampler proposes a sample zk that has been already visited

by the algorithm, the next sample is selected randomly

satisfying pðx; yÞ[ l; where l is the mean value of pðx; yÞcomputed for all the pixels of the image.

Given a candidate pixel proposed by the sampling

technique, the process arrives to the line segment genera-

tion stage. It starts with the point refinement module, which

uses mean shift (MS) procedures on a multidimensional

space composed of the position of the pixels and their

dominant local orientation, denoted as xk ¼ ðxk; yk; hkÞ1.

The MS searches for a local maxima on this space, i.e., it

looks for the pixel in the neighborhood of the candidate

pixel that most likely belongs to a line segment.

From this local maxima, an efficient line growing

strategy, based on the Bresenham algorithm is applied to

connect pixels until the end points of the line segment are

reached. This scheme is applied iteratively to enhance the

accuracy of the generated line segments. All the pixels

swept by the generated line segment are marked as visited,

and thus not available to be selected anymore during the

sequential sampling stage.

The overall result of the strategy is that the sequential

selection of candidates with the slice sampler is much more

efficient than processing the whole image in search for line

segments (as done by most works based on pixel clustering

techniques), and even more than randomly selecting points

on the image without any guiding criteria (as the works

about random sampling on the Hough domain). On the

other hand, the accuracy on the line segment-fitting process

is provided by the MS procedure that is used to refine the

candidates and generate good starting and ending points for

the growing algorithm.

4 Sequential sampling strategy

In this section we describe the sequential sampling stage,

which is based on a general sampling algorithm: the slice

sampler. Also, the computation of the likelihood function

for line segments and the initialization and termination

processes are presented.

Sobel and Eigendecomposition

Computation of (µ1,µ2) and µ

Get sample p(x,y) > µ(jump)

Terminate?

End

Slice sampler 2D

Visited?

Point refinement(wMS)

Iterative line growing(Bresenham + wMS)

Update visited

Sequential samplingOperations at pixel level Line segment generation

p(x,y)

Image

zk

YES

NO

zk+1

YES

NO

xk+1^

^

Fig. 1 Flow chart of the proposed strategy. The image is processed in

order to obtain the likelihood distribution pðx; yÞ defined by the

parameters (l1, l2). The mean value of pðx; yÞ;l; is used to generate

the first sample of the algorithm and start running the slice sampler,

which sequentially generates samples, zk. The last stage is the line

segment generation, in which the samples are refined and used as

starting point for the iterative line growing algorithm that finally

generates the line segment

1 For the sake of clarity in the notation, italic characters correspond

to scalar values while bold characters represent arrays.

Pattern Anal Applic (2011) 14:149–163 151

123

4.1 Slice sampling

The slice sampler [25] is a general sampling strategy born

in the field of inference methods based on numerical

sampling (typically known as Markov Chain Monte Carlo

techniques) [4]. It allows to sequentially obtain samples,

fzkgNk¼1 from a arbitrary target pdf, pðzÞ: The only

requirement to apply this algorithm is that the value pðzÞshall be evaluated for any given value of z:

As described in [4], the slice sampling improves the

results, in terms of efficiency, of typical sampling

approaches based on the Metropolis–Hastings (MH) algo-

rithm [12]. This algorithm (MH) has an important draw-

back that makes it inefficient for the proposed line segment

detector as it is sensible to the step size, given by the

proposal distribution. If it is chosen too small, the process

behaves as a random walk, which makes the algorithm

converge very slowly and, on the contrary, if it is too large,

the rejection rate may be very high, hence not achieving

accurate results. The advantage of the slice sampler is due

to its ability to automatically adapt its step size according

to the characteristics of the pdf.

For a better understanding of the slice sampler, let us

first consider the univariate case: p(z). Slice sampling

works by augmenting z with an auxiliary random variable

u and then sample from the joint (z, u) space [25]. Given

the previous sample zk-1, u is uniformly drawn in the range

[0,p(zk-1)]. Fixed u, the sample zk is obtained from the

‘‘slice’’ through the distribution defined by {z:p(z) [ u}.

This criterion is illustrated in Fig. 2a. Nevertheless, it is

difficult to find the limits of the slice and thus to draw a

sample from it. For that reason an approximation is done

by means of creating a quantized local slice, delimited by

z0 and z1 as shown in Fig. 2b. To obtain these limits, the

value p(z) is evaluated at left and right of zk-1 using fixed

length steps (the quantification step) until p(z) \ u. The

next sample, zk, is obtained by uniformly sampling on this

range (iteratively until p(zk) [ u).

For the line segment detection, the sampling has to be

carried out on a two-dimensional space. We propose to

obtain samples by sequentially applying one-dimensional

slice sampling at each dimension, x and y. Figure 3 illustrates

this procedure, depicting the contour plot of a zoom on a two-

dimensional, pðzÞ; function. For a given sample zk�1 ¼ðxk�1; yk�1Þ; depicted as a black square, the one-dimensional

slice criterion is applied first on x: yk-1 is fixed and a new xk is

delivered. This value is then fixed (represented as a circle)

and the one-dimensional procedure is repeated for y. The

result is the new two-dimensional sample zk ¼ ðxk; ykÞ:Hence, the 2D slice sampler used is a combination of two 1D-

slice steps, which results in a fast an efficient 2D sampling.

As a remark, note that if the order of the one-dimensional

samplers is reversed, a different final sample zk could be

generated, though this is irrelevant for the line segment

generation, as the mean shift procedures refine these samples

considering a similar neighborhood.

4.2 Line segment likelihood

The slice sampling algorithm requires the construction of a

target pdf, which is, in this case, the likelihood of the image

pixels to belong to line segments.

(a)

(b)

Fig. 2 Univariate slice sampling: a the uniform u value determines

the slice through p(z); and b the practical implementation uses fixed

length steps to determine the range in which z is uniformly sampled

,

,

Fig. 3 The level set plot allows to observe that zk-1 is located in a

region with levels between 0.6 and 0.8. The slice procedure on

x produces the region limited by x0 and x1. The selected sample xk is

fixed and the region between y0 and y1 is obtained for the y dimension.

The final sample is zk = (xk, yk)


123

Different methods have been addressed in the literature

to characterize image pixels to belong to corners, homo-

geneous regions or line segments. Most of them use the

covariance matrix or image tensor matrix for each pixel

and their associated eigenvalues, (k1, k2) with k1 [ k2, and

eigenvectors, ðe1; e2Þ (details can be found in works as

[33]).

The eigenvalues illustrate the dispersion of the gradient

along their associated eigenvectors [28] as can be seen in

Fig. 4. For that reason, a point that belongs to a line seg-

ment is described by a significant value of k1, while k2 is

close to zero. Analogously, a corner is given by a pair of

eigenvalues with similar magnitude both being far from

zero, whereas pixels inside homogeneous regions have

both eigenvalues close to zero. These cases are depicted in

Fig. 4, where the obtained eigenvalues are represented as

the axis of the ellipse fitting the gradient distribution of the

neighborhood of the considered pixel.

In this work, we propose to handle the eigenvalues

information by means of a novel function composition,

based on the eigenvalues of the pixels of the whole image

to generate a function that maps back the likelihood value

of a pixel given its particular eigenvalues. The function

satisfies two criteria: on the one hand, it returns high values

only for pairs of eigenvalues for which one of them is much

higher than the other; and, on the other hand, it decreases

rapidly when the eigenvalues are both high or low. The

function is defined as:

p ¼ g � f : I ! ðk1; k2Þ ! R

ðx; yÞ 7! pðx; yÞ ¼ gðf ðx; yÞÞ ð1Þ

where f(x, y) is the function that gives, for each pixel

(x, y), the pair of corresponding eigenvalues (k1, k2), and

g(k1, k2) is the function that determines the likelihood of a

pair of eigenvalues to correspond to a line segment, defined

as:

gðk1; k2Þ ¼ ð1� expð�k1=l1ÞÞ expð�k2=l2Þ ð2Þ

where the parameters (l1, l2) are, respectively, the average

values of the eigenvalues computed over the whole image,

and have been obtained in the first stage of the proposed

strategy. The function g(k1, k2) is the product of two

independent functions on each eigenvalue. The first term,

on k1, penalizes the response for those pixels having their

largest eigenvalue too small, i.e., those that likely belong

to an homogeneous region. The second term, on k2,

enhances the response for those pixels with a very low

smallest eigenvalue, thus penalizing those corresponding

to corners, which are described by high values of k1/l1

and k2/l2. The result is a good representation of the

likelihood of a pixel to belong to a line segment. This

function shows high response values for pixels with a

significant largest eigenvalue, and a very low smallest

one. For one example image, this function is illustrated in

Fig. 5a, while the associated scatter plot of (k1, k2) is

shown in Fig. 5b.

As an example, the pðx; yÞ values of all the pixels in an

image have been computed. Fig. 6 shows, in (a) the ori-

ginal image, which contains large homogeneous regions,

significant line segments and corners; (b) shows an scaled

representation of the line segment likelihood, where pixels

with higher intensity have higher probability to belong to

line segments.

Ix

Iy

λ1 λ2>>

(a)

Ix

Iy

λ1 λ2

(b)

Ix

Iy

λ1 λ2 0

(c)

≈ ≈ ≈

Fig. 4 Eigenvalues illustrating gradient structures. Black dots are the

(Ix, Iy) values of the neighbor pixels of an example pixel that

corresponds to: a a line segment, b a corner or an heterogeneous

gradient region, and c homogeneous region

0 0.01 0.02 0.03 0.04 0.050

1

2x 10

−4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(a)

(b)

Fig. 5 The level sets of function g(k1, k2) shown in a. In b a scatter

plot of the set of pairs (k1, k2) for an example image. The

concentration at the origin is typically due to the high number of

pixels that belong to homogeneous regions of the image


123

4.3 Initialization and termination

The initialization of the algorithm is done by the iterative

selection of pixels uniformly along the image until a pixel

is found with pðx; yÞ[ l; where l is the mean value of

pðx; yÞ computed on the whole image. This initialization

ensures a good candidate point for the line generator stage

of the algorithm. The reason is that the histogram of pðx; yÞtypically follows a decreasing exponential distribution.

Hence, most pixels of the image do not belong to line

segments and form a mode close to zero. So, the selection

of starting points above the mean ensures that we are not

selecting points that do not likely belong to line segments.

In the next steps, the slice sampler acts sequentially, find-

ing automatically pixels with pðx; yÞ similar to the one of

the initialization. Each time a line segment is generated, all

the pixels that belong to it are removed from the set of

pixels that can be selected by the slice sampler, as well as

their neighbors in a window of r 9 r pixels, where r is the

spatial bandwidth of MS; more details are given in next

section.

A major virtue of this strategy is that typically only one

candidate is required to generate a whole line segment,

which marks as ‘‘visited’’ all its pixels so that the sampling

strategy never draws a sample for that segment again. For

that reason, it may happen that the slice sampler finds out

that all the pixels surrounding the previous sample have

been marked as visited, so that no more segments need to

be detected in that region of the image. Therefore, the

initialization criterion is applied again to produce a new

start for the slice sampler, which will be referred to as a

jump. Hence, the sampler allows to jump naturally from

one region to another in the image according to the line

segments generated in previous steps.

The proposed scheme sequentially selects the pixels of

the image in order of importance, and allows to sweep all

the pixels till the end. Nevertheless, it is also possible to

determine a termination criteria according to the available

computational resources, for example stopping when a

maximum number of jumps is reached.

In summary, the sampling strategy sequentially generates

samples zk ¼ ðxk; ykÞ; with an associated orientation hk ¼arctanðIyðxk; ykÞ=Ixðxk; ykÞÞ: These samples are used by the

line segment generation step to generate the line segments.

5 Line segment generation

For this stage of the algorithm, the sample zk and its

associated normal orientation hk are used to compose the

vector, or edge-point, xk = (xk, yk, hk), which will guide

the line segment generation process. This process is com-

posed of three independent stages, as shown in Fig. 1,

which are applied on each new candidate xk. The point

refinement step applies a multidimensional weighted mean

shift procedure. It starts on xk and searches for a better

representative edge-point of the line segment, denoted as

xk ¼ ðx; y; hkÞ. The line growing step searches for the two

end points of the line segment under analysis. It is based on

an oriented local growing algorithm that starts on zk gov-

erned by hk. The final step marks as visited those pixels

that compose the line segment.

5.1 Refinement of the sample

The probabilistic nature of the slice sampling process

delivers samples, xk; with high value of p(x, y), that likely

belong to a line segment. Nevertheless, in their neighbor-

hoods there might be edge-points that better represent the

position and orientation of the line segment. The refine-

ment step searches for this representative, denoted as xk;

using a novel multidimensional wMS procedure, inspired

on the well-known mean shift algorithm [8, 15]. MS is an

iterative, non-parametric algorithm that can be used to find

local maxima of an unknown density function from which

a set of samples is available. At each iteration, MS operates

on an starting sample and moves towards the most dense

region of the search space in the neighborhood.

We propose to use MS with data samples weighted

according to their likelihood values. For a given sample xk;

Fig. 6 The original image in shown in a, and the scaled pdf map in b.

Note that corners receive low likelihood values and thus are depicted

in black, as well as homogeneous regions. Only line segments have

high likelihood values, painted in high bright levels; c shows a zoom

of the highlighted region (located at the bottom right part of the image

shown in b). In c, the sample xk and the refined sample xk are shown.

The dotted vector represents an intermediate iteration of wMS; finally

d is the 3D representation of pðx; yÞ values of the considered region


123

its associated weighting factor is xk ¼ pðx; yÞ: Hence, the

wMS procedures find maxima following a twofold target.

On the one hand, it looks for the maximum of pðx; yÞ within

the neighborhood of zk; which is possible thanks to the

weighting factors applied to the positions (x, y), which are

distributed uniformly along the image. The association of a

weight to each pixel allows wMS to handle this spatial

information as a non-uniform distribution. On the other

hand, the orientation component of the search space makes

that the wMS procedures move towards edge-points highly

homogeneous in their orientations, which needs not to be at

the maximum of pðx; yÞ in the neighborhood, but a better xk

from which start the line growing stage.

Figure 6d shows a 3D representation of these values,

corresponding to the zoomed region shown in (c): the

elevation associated to each pixel corresponds to its line

segment likelihood level, which is the weight applied

during the wMS procedures. As observed, the pixels that

belong to the line segment have higher weights, which are

the target of the wMS procedures.

Mean shift works as a kernel-based non-parametric

estimator of the pdf from which samples xk are drawn. Let

us consider a set of samples, fxigi¼1...N ; from an unknown

density f ðxÞ: The multivariate weighted kernel estimator of

f ðxÞ is defined as:

bfxðxÞ ¼1

PNi¼1 xi

XN

i¼1

xiKHðx� xiÞ ð3Þ

with the kernel defined as:

KHðxÞ ¼1

jHj12

Kx

H12

� �

ð4Þ

where H is a symmetric positive definite 3 9 3 bandwidth

matrix that specifies the ‘‘width‘‘ of the kernel at each

dimension. As a fully parameterized H increases the

complexity of the estimation [8], the bandwidth matrix H is

chosen for our approach as a diagonal matrix containing

the corresponding bandwidths for each dimension: H ¼diag½h2

x ; h2y ; h

2h�:K is obtained from the product of

symmetric univariate kernels:

KðxÞ ¼Y

3

d¼1

cdkðxdÞ ð5Þ

where xd are each component of the vector x; cd are the

normalization constants, and the function k is defined as the

Epanechnikov kernel [8]. Substituting (5) into (4), and the

result into (3) yields:

bfxðxÞ ¼jHj�

12C

PNi¼1 xi

XN

i¼1

xi

Y3

d¼1

kxd � xi;d

hd

� �2 !

ð6Þ

where xi,d is the dth component of data sample xi and hd is

its corresponding bandwidth. Note that, as opposed to most

approaches using MS, we are describing an estimator that

considers not only a set of weighting factors for the kernels

but also a multidimensional bandwidth matrix, which

contains different bandwidth values for each dimension

[15].

The modes in f ðxÞ that we are interested in, are located

at the zeros of the gradient, rf ðxÞ: An estimator of the

gradient of f ðxÞ is the gradient of bfxðxÞ; that can be rep-

resented as follows:

rbfxðxÞ ¼ AX

N

i¼1

xiGðx� xiÞ" #

�X

N

i¼1

xiGðx� xiÞ !�1

XN

i¼1

xiGðx� xiÞxi

!

� x

2

4

3

5 ð7Þ

where A is a matrix gathering the constants and the

bandwidth matrix:

A ¼ 2jHj�12C

PNi¼1 xi

H�1 ð8Þ

and GðxÞ is a diagonal 3 9 3 matrix defined as:

GðxÞ ¼ diag �k0x2

h2x

� �

;�k0y2

h2y

!

;�k0h2

h2h

� �

" #

ð9Þ

The function k0 is obtained as the differentiation of the

kernel k, as defined in (5).

The last bracket in (7) is the mean shift vector, which

represents the difference between the weighted mean of the

data samples and the center of the kernel:

mwðxÞ ¼PN

i¼1 xiGðx� xiÞxi

� �

� xPN

i¼1 xiGðx� xiÞð10Þ

The application of the wMS procedure generates mean

shift vectors at each iteration towards the refinement of xk:

An example is shown in Fig. 6c. The candidate vector-

point given by the slice sampler is clearly near to a line

segment, but the refined version, xk; is much more accurate

in position and orientation.

5.2 Line growing algorithm

The refined sample, xk ¼ ðxk; yk; hkÞ; is used as a starting

point for the generation of the line segment. The line

segment is delimited by its end points, za ¼ ðxa; yaÞ and

zb ¼ ðxb; ybÞ; which are found using an oriented growing

strategy inspired on the Bresenham algorithm [5]. This

method is illustrated with the example upper line segment

of Fig. 7a: starting from the edge-point ðxk; ykÞ;


123

corresponding to the arrow, the pixels along the orientation

hk (shown as dots) are connected while their orientations

are in the range hk � Dh: In our experiments we have found

that a good value for this parameter is Dh ¼ p8; which

seems to work fine for a wide range of types of images, and

it is also proposed by other authors that work with CCA

techniques [7, 13]. The extreme points (marked with

squares in the figure) of the obtained cluster are the target

end-points of the line segment.

This procedure is applied iteratively until convergence is

reached. For this purpose, an error measurement is defined

for the set of points, indexed by i, covered by the line

segment at iteration j, computed as follows:

�i ¼1

M

XM

i¼1

jhi � hjj ð11Þ

where hi is the orientation of each point of the line seg-

ment, hj is the orientation that guides the line growing

algorithm and M is the number of pixels connected by the

growing algorithm. Note that the maximum error at each

pixel is j hi � hj j �Dh (where Dh ¼ p8), hence 0� �i�Dh:

The process is considered to have converged when

�j �j�1; i.e., when the new iteration returns a set of pixels

that does not fit better to the growing direction hj than the

previous iteration. The value of hj for each new iteration is

obtained as the orientation of the segment that joins the

refined end-points of the previous iteration. These points

are refined as described in Sect. 5.1 by means of the

application of one wMS procedure to each of them. The

process is guaranteed to converge since the number of

candidate end-points for a line segment is finite, and so are

the orientations hj; so that there is a global minimum error

�j: Nevertheless, we have seen in our experiments that, in

average, the process converges in two or three iterations.

The reason of using this iterative strategy is to com-

pensate potential small deviations between hk and the

actual line segment orientation. An example is depicted in

the lower segment of Fig. 7a. As shown, the application of

two new wMS procedures, one on each end-point restarts

the growing algorithm to find a more accurate line seg-

ment, achieving convergence at the second iteration.

Fig. 7b shows the obtained segments for the examples

shown in (a).

6 Complexity of the algorithm

In this section we provide an estimate of the complexity of

the whole algorithm. For this purpose we would assume

that sums, products, divisions and the rest of basic opera-

tions are equally costly (as done by other authors of related

works [14]).

The number of operations of the proposed method is

approximately proportional to the number of pixels of the

image, N. The most time consuming part of the algorithm is

the computation of the estimate of the likelihood function

pðx; yÞ; since it involves the calculation of the eigenvalues

and vectors for all the pixels of the image. The operations

that are carried out are the calculation of the Sobel

approximation to the spatial derivatives O(16N), the com-

putation of the gradient orientation O(2N), the obtention of

the eigenvalues O(10N) and the computation of l, O(N).

The result is O(29N) which is proportional to the size of the

image.

The rest of computations are done for each line segment

and consume much less processing resources. The Bre-

senham algorithm plus the wMS-based refinement is

O(L ? 3R), where L is the average length of a line seg-

ment, and R = r 9 r is the squared spatial bandwidth of

wMS (r = hx = hy). O(L) refers to the computation of the

position of each candidate pixel in the growing strategy,

and O(3R) corresponds to the execution of 3 MS proce-

dures, one for setting the starting point of the growing

strategy, and one for each end-point to refine its position.

Note that M(L ? 3R) 29N for average values of L = 50

pixels, M = 400 line segments and N = 600 9 400 pixels.

Fig. 7 Oriented growing

algorithm: a performance for

two different growing cases, one

of them with an accurate

starting orientation (top), and

the other (bottom) with a

refinement step that corrects the

orientation; b the obtained line

segments for both cases are

correct


123

This theoretical analysis is reflected in the results of next

sections, especially in Table 1, where the processing time

of the algorithm is evaluated for the detection of different

number of line segments.

7 Tests and results

This section includes the description of different tests

applied on the developed SSWMS algorithm, which are

focused on demonstrating its performance in terms of

speed, accuracy, and flexibility. A comparison with other

related state-of-the-art line detectors is also presented.

7.1 Performance analysis

For the tests, we have implemented our solution in C??

programming language using OpenCV v1.1 libraries, and

we have run the tests on a Core2Duo processor at 2.2 GHz.

We have collected and processed a large set of images of

several environments, such as buildings, roads, facades,

portraits, etc., with different formats, sizes and levels of

noise. Figure 8 shows some of these images and their

associated detected line segments. Observe that most of

significant line segments in the images are correctly

detected, achieving great accuracy and resulting in a very

reduced number of false negatives or missdetections. Fur-

thermore, it is also remarkable that the algorithm generates

very few false positives in areas of the image that contain

edges but without actual line segments, such as the leafs of

the trees, the mountains, the hair or the small elements at

the far distance. Ordered from left to right and from top to

bottom, the third image shows a very interesting property

of the SSWMS thanks to the sequential sampling step: it

provides satisfactory results for all the straight edges of the

image even though there are some (the ones corresponding

to the shadows) that are much stronger than the rest of

edges in the image. This would cause problems in methods

that process sequentially based on sorting the edge points

of the image according to their magnitude.

Regarding processing times, the algorithm was designed

to perform in real time for medium size images, achieving

an average of 125 ms (8 fps) for images about 640 9 480

pixels and below 50 ms (20 fps) for 360 9 288 pixels,

which are typical image sizes for most online video pro-

cessing applications. The processing time increases linearly

with the size of the images, so that larger ones, with HD

resolutions such as 1,280 9 720, 1,440 9 1,080 or

1,920 9 1,080 require processing times of 430 ms (2.32

fps), 650 (1.5 fps), and 1,000 ms (1 fps), respectively.

7.2 Discussion about parametrization

It must be considered that the processing times presented

before correspond to the execution of the SSWMS until it

has searched for line segments along the whole image. The

sequential and probabilistic nature of the algorithm allows

to stop computing when a requested number of line seg-

ments has been obtained. Typically, for most computer

vision applications, only the most representative line seg-

ments are of utility. Table 1 shows the time consumed by

the SSWMS for the detection of different number of line

segments for different image sizes. As shown, the pro-

cessing times are very reduced, and perfectly suitable for

real-time computer vision applications. On the one hand,

the computational time required for images with resolu-

tions below 640 9 480 do not significantly vary for dif-

ferent target numbers of line segments, as the most

consuming part of the algorithm is the computation of the

eigenvectors. Therefore, for medium and small images, it is

worthy to apply the full search, with no parameters, as the

gain of time is negligible. On the other hand, for larger

image sizes, the reduction of time when reducing the

number of requested line segments is more significant. For

instance, the application of the SSWMS on an image with

1,600 9 1,200 pixels, would spend near 1,000 ms, while

applying the SSWMS limiting the search to the first 500

line segments, would result in a reduction of processing

time in about 30%. In any case, detected line segments are

those ones having more intense values of p(x, y), which are

typically the most significant lines in the image.

The only parameter of the algorithm that has to be fixed,

as mentioned along the paper, is the spatial component of

the bandwidth vector of the mean shift procedures, r. By

default it is fixed to r = 3, which makes the system per-

form excellent for medium-size images. Nevertheless, an

even better performance can be obtained by automatically

adapting this value according to the size of the images. The

Table 1 Average process time (in milliseconds) for different image

sizes and requested line segments

Image Size Requested number of line segments

50 100 500 1

180 9 144 16 16 16 16

320 9 240 31 31 31 31

360 9 288 46 47 47 47

640 9 480 125 125 156 156

1,024 9 720 281 281 313 390

1,366 9 768 391 391 422 563

1,280 9 1,024 485 484 531 688

1,680 9 1,050 641 656 687 922

1,600 9 1,200 703 703 735 1,000

1,900 9 1,200 812 844 846 1,172

3,072 9 2,304 2,578 2,578 2,609 3,781


123

bandwidth is an indicator of the resolution of the detector,

i.e., the minimum distance between two detected line

segments. For that reason, as the images increase in size

better results are obtained increasing the size of this

parameter. The results for an example image using differ-

ent values of r is shown in Fig. 9. As shown, for small

values of r, the number of detected line segments is much

higher. Increasing this parameter makes the system work

towards the detection of dominant line segments. For

instance, for r = 9, the number of detected line segments is

75, which mostly correspond to the main edges of the

building.

7.3 Comparison with other methods

We have selected two line segment detection algorithms to

compare with our method: the LSD (Line Segment Detector

[13]) and the PPHT (Progressive Probabilistic Hough

Transform [11]) for which efficient implementations are

available (the authors of LSD offer their implementation in

their website, while the PPHT is a very well-known algo-

rithm that has been efficiently implemented as a function

inside the OpenCV 1.1 libraries). The motivation of this

selection is that this implementation of the PPHT is a very

good representative of Hough-based methods as well as, in

our knowledge, it is the fastest method in the literature. The

LSD has been recently published and it offers linear-time

operation and very accurate results, and is selected for

comparison as it is the best algorithm derived from the

Burns method [6].

The first test was conducted to check the performance of

these algorithms in the presence of noise. An example

image, with significant added noise and the detected line

segments obtained with the different methods is shown in

Fig. 10. The upper row of this figure illustrates the

robustness of the SSWMS against noise: the LSD is unable

to detect segments, the PPHT produces hundreds of noisy

line segments, while our approach detects, partially split-

ted, the most important line segments in the image. In the

second row, a gaussian filter is used to remove partially the

noise, as suggested by [13] to improve its accuracy in

the presence of noise. In this situation, both SSWMS and

LSD show similar performance, while the PPHT still

generates too many noisy segments.

Figure 11 shows a scatterplot of the processing time of

our approach against the size of the image (in pixels)

compared to the PPHT and LSD algorithms: (a) shows the

comparison for small and medium size images, and (b) for

Fig. 8 Some results of the SSWMS applied on different real images


123

large images. The PPHT is on average 10% faster than

SSWMS considering the whole set of images used for the

comparisons (more than 200 images), while the LSD is

between 10% (for large images) and 30% (for small and

medium size images) slower than our approach. Note that

for the comparison we had to tune the parameters of the

PPHT for each image independently to obtain the best

results in terms of speed and accuracy (such as the

resolution of the transform space, the minimum vote

threshold, and minimum length of the line segments), while

the SSWMS needs no parameter tuning. The accuracy of

these methods is illustrated, for different images in Fig. 12.

As shown, the PPHT shows the higher number of false

detections and missdetections. The LSD offers very good

results for almost all images, specially those that have

high-contrast edges. Nevertheless, for some images, such

Fig. 9 Results of the SSWMS

line segment detection using

different values of the mean

shift spatial component of the

bandwidth, r

Fig. 10 Comparison of the

performance of the compared

methods against noise. The

upper row shows the results

obtained from the noisy image;

and the second row shows the

results for the smoothed version

of the image


123

as the one showing the ground (first row in Fig. 12), it fails

to detect important line segments, and it produces too many

line segments, retrieving multiple replied segments that

add no information about the scene. In all these cases, the

SSWMS shows excellent performance, detecting the most

important line segments with a hghly reduced number of

false detections and missdetections.

7.4 Recall and precision

This section describes the comparison of the performance

of the proposed algorithm with other methods in terms of

recall and precision. Recall is related to the number of

missdetections, and precision to the number of false

alarms. These evaluation metrics can be computed as:

Recall (%) ¼ # Correct detections

# Ground truth line segmentsð12Þ

Precision (%) ¼ # Correct detections

# Total detectionsð13Þ

In this context, a detected line segment is considered to

be a correct detection if it is similar to an existing ground

truth line segment in terms of relative orientation, distance

between mid-points and length. The target of the test is to

evaluate, with objective results, the detection results on

different situations, which correspond to typical conditions

of real images: (1) variable size, (2) noise, and (3)

perspective distortion. Modifying the size of the images

helps to evaluate the response of the methods against

variations of the length of the line segments. The

introduction of noise shows how flexible the algorithms

are in providing good results when the spatial coherence of

pixel information (such as gradients) is lost or corrupted

due to the noise. Finally, the perspective distortion

illustrates another important property: this type of

transformation implies a non-homogeneous modification

of the image area. Hence, the length of the line segments is

non-uniformly modified. Therefore, this test illustrates the

ability of the methods to compute long and short line

segments in the same image.

The RP values have been obtained for an example image,

shown in Fig. 13a, and for a number of its modifications,

illustrated in Fig. 13b, c, which follow the above-mentioned

variations in size, noise and perspective distortion. The

recall and precision results are shown in Fig. 14, separately

for the three types of modifications, where the values cor-

responding to the original image are shown with a thick

black marker in the three graphs2. The graph showing

‘‘variable size’’ depicts the RP values obtained increasing

the size of the image; ‘‘variable noise’’ stands for the

addition of Gaussian noise with standard deviation values

ranging from 0 to 0.07 in 0.01 steps. An example image

with noise is shown in Fig. 13b. Finally, ‘‘variable per-

spective distortion’’ illustrates the results obtained by

applying perspective transformations on the image, such as

the one shown in Fig. 13c.

The results related to the modification of the size of the

image indicates that the SSWMS renders very accurate

results with very low missdetections and false alarms,

while the performance of the LSD decreases as it provides

more false alarms as the size of the image increases. The

recall of the three methods increases since it is easier to

detect all the existing line segments as the image increases.

The addition of noise, as already commented in the

example of Fig. 10, is very harmful for all methods.

Therefore, RP curves tend to decrease both the recall and

precision values, although to a lesser extent for the

SSWMS algorithm, which still shows RP above 50/50 for

added noise with 0.05 standard deviation.

The results corresponding to the images with perspective

distortion show that the recall values decrease as the number

0

100

200

300

400

500

600

700

800

0 100000 200000 300000 400000 500000 600000

Image size (pixels)

LSD PPHT SSWMS

Linear (LSD) Linear (PPHT) Linear (SSWMS)

0

2000

4000

6000

8000

10000

12000

14000

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000

Image size (pixels)

LSD PPHT SSWMS

Linear (LSD) Linear (PPHT) Linear (SSWMS)

(a)

(b)

Fig. 11 Process time of the SSWMS, PPHT and LSD methods for more

than 200 images with different sizes: a images up to about 600 9 900

pixels; and b larger images up to 3,702 9 2,304 pixels. The linesrepresent the linear tendency of the data along the image size axis (solidlines represents our method). As shown, the faster method is the PPHT,

while the SSWMS shows higher performance than LSD in both cases

2 SSWMS RP, 91.73/95.35; LSD RP, 90.35/73.33; PPHT, RP: 38.19/

89.91. These numbers mean that both the LSD and SSWMS offer

great results for this image, although the LSD delivers more false

alarms. The PPHT suffer more missdetections and thus its recall value

is very low.


123

of line segments whose length is reduced grows. Nevertheless,

the SSWMS algorithm still keeps a very good value of RP

especially compared with the PPHT, which is not able to

detect a large number of line segments due to the distortion.

8 Example application

As commented in Sect. 1, line segments are useful image

features for a number of applications in the computer

Fig. 12 Comparisons of the

line segments obtained for

different images applying

different methods. From left to

right: SSWMS, LSD and PPHT

Fig. 13 Example image used to

test recall and precision: a origi-

nal image with ground truth line

segments; b image with added

gaussian noise; and c image

with perspective distortion

0

20

40

60

80

100

0 50 100

Pre

cisi

on

(%

)

Recall (%)

Variable size

0

20

40

60

80

100

0 50 100

Pre

cisi

on

(%

)

Recall (%)

Variable noise

0

20

40

60

80

100

0 50 100

Pre

cisi

on

(%

)

Recall (%)

LSD PPHT SSWMS

Fig. 14 Recall and precision

graphs for different situations:

varying size, adding noise and

perspective distortion for the

three methods. The starting

point is marked with a thick

black edge, and is the same for

the three graphs


123

vision field. As an example, we have successfully used the

proposed SSWMS algorithm for the computation of van-

ishing points in sequences of images. The accuracy of the

SSWMS has been evaluated computing the orientation

error of the obtained line segments with respect vanishing

points following the approach by Kosecka and Zhang [20].

For this purpose we have used the York Urban Data

Base (YUDB) [10], which is a database of 102 images of

man-made structured environments with ground truth

vanishing points (which correspond to the three main

orthogonal directions of the scene). For each image of the

YUDB we have run the SSWMS and we have evaluated

the orientation error of the obtained line segments with

respect to each vanishing point. Figure 15a shows an

example histogram of the error of the detected line seg-

ments with respect to one vanishing point. The mode

around zero correspond to the error of the line segments

actually meeting at the vanishing point. The rest of the

histogram corresponds to line segments not meeting that

vanishing point. As shown, the peak can be fitted as a

normal distribution with a mean close to zero and a low

variance. The error of the line segments obtained with the

SSWMS is compared with the actual error of the image

information, which can be characterized by the gradient

vectors computed for each pixel of the image. The corre-

sponding histogram of these gradient vectors is shown in

Fig. 15b. As can be observed, the error distribution is more

peaked for the line segments, which illustrates that the use

of SSWMS reduces the error in the orientation and thus

increases the accuracy of the information that is used to

compute vanishing points. This experiment is repeated for

all the vanishing points of all the images of the YUDB,

obtaining mean and standard deviation values. The result is

illustrated in Fig. 16, which shows a scatter plot of the

obtained statistics, showing with dots the results for the

SSWMS line segments, and with crosses the results of

the gradient-pixels. As can be observed, the distribution of

error of the SSWMS is more concentrated on zero mean

values, with lower standard deviation.

9 Conclusions

In this paper, we have proposed a novel approach for

accurate and fast detection of line segments in images,

specially devoted for real-time vision applications. It has

been designed to adapt itself to the characteristics of the

images; hence, there is no need to tune any input param-

eter. It is based on a sequential sampling strategy, which

uses the slice sampler to draw pixels in the image that

likely belong to line segments. At each drawn sample, an

iterative line growing strategy, based on mean shift is

applied which finally finds the end-points of the line

segment.

The outstanding performance of this strategy has been

demonstrated in terms of accuracy, flexibility and, mainly,

speed. A comparison with other line segment detection

methods has been carried out. It has shown that our method

offers an excellent trade-off between very accurate meth-

ods and fast ones. One of the major abilities of the pro-

posed algorithm is that it finds the most important line

segments for any type of images, including noisy ones,

without the need of tuning any parameter.

Acknowledgments This work has been partially supported by the

Ministerio de Ciencia e Innovacion of the Spanish Government under

project TEC2007-67764 (SmartVision), and by the Comunidad de

Madrid under project S-0505/TIC-0223 (Pro-Multidis).

−1.5 −1 −0.5 0 0.5 1 1.50

10

20

30

Error angle (rads)

SSWMS − Error angle histogram

−1.5 −1 −0.5 0 0.5 1 1.50

500

1000

1500

2000

Error angle (rads)

Gradient−pixels − Error angle histogram

(a)

(b)

Fig. 15 Error angle histograms: a line segments obtained with

SSWMS; and b gradient-pixels

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.1

−0.05

0

0.05

0.1

Standard deviation (rad)

Mea

n(ra

d)

Fig. 16 Scatter plot of the mean and standard deviation of the normal

fit of the inliers mode of the error histogram. The results of the normal

fit of the line segments obtained with SSWMS is shown in dots. The

normal fit corresponding to gradient-pixels is shown with crosses.

Realize that there are 306 data points, since each image of the YUDB

contains three vanishing points


123

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Line segment detection using weighted mean shift procedures on a 2D slice sampling strategy

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